A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius (^(@)C) can be modeled with the expression 23+24*2^(-0.014 t), where the variable t represents the time in minutes. Approximately how many minutes will it take for the temperature to reach 29^(@)C ? (Round to the nearest minute.)

Question

A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius  (^(@)C) can be modeled with the expression  23+24*2^(-0.014 t), where the variable  t represents the time in minutes. Approximately how many minutes will it take for the temperature to reach  29^(@)C ? (Round to the nearest minute.)

Bytelearn · Accepted Answer

Write temperature model and target: Write down the given temperature model and the target temperature. The temperature model is given by the expression 23 + 24 * 2^(-0.014t), and we want to find the time t when the temperature reaches 29 degrees Celsius.Set model equal to target: Set the temperature model equal to the target temperature and solve for t. 29 = 23 + 24 * 2^(-0.014t)Isolate exponential term: Subtract 23 from both sides to isolate the exponential term. 29 - 23 = 24 * 2^(-0.014t) 6 = 24 * 2^(-0.014t)Solve for exponential part: Divide both sides by 24 to solve for the exponential part. 6 / 24 = 2^(-0.014t) 1/4 = 2^(-0.014t)Set exponents equal: Recognize that 1/4 is a power of 2, specifically 2^(-2), and set the exponents equal to each other to solve for t. 2^(-2) = 2^(-0.014t) -2 = -0.014tSolve for t: Divide both sides by -0.014 to solve for t. t = -2 / -0.014 t ≈ 142.857Round to nearest minute: Round the answer to the nearest minute. t ≈ 143 minutes

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